Theorem intidOLD 5458

Description: Obsolete version of intidg 5457 as of 18-Jan-2025. (Contributed by NM, 5-Jun-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
intid.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
intidOLD	⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem intidOLD

Step	Hyp	Ref	Expression
1		snex 5431	. . 3 ⊢ {𝐴} ∈ V
2		eleq2 2823	. . . 4 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
3		intid.1	. . . . 5 ⊢ 𝐴 ∈ V
4	3	snid 4664	. . . 4 ⊢ 𝐴 ∈ {𝐴}
5	2, 4	intmin3 4980	. . 3 ⊢ ({𝐴} ∈ V → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴})
6	1, 5	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ⊆ {𝐴}
7	3	elintab 4962	. . . 4 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥(𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥))
8		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
9	7, 8	mpgbir 1802	. . 3 ⊢ 𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
10		snssi 4811	. . 3 ⊢ (𝐴 ∈ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} → {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥})
11	9, 10	ax-mp 5	. 2 ⊢ {𝐴} ⊆ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥}
12	6, 11	eqssi 3998	1 ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {cab 2710  Vcvv 3475   ⊆ wss 3948  {csn 4628  ∩ cint 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-int 4951

This theorem is referenced by: (None)